Elliptic functional differential equation with affine transformations

J. Math. Anal. Appl. 480 (2019) 123403

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Elliptic functional differential equation with affine transformations ✩ L.E. Rossovskii a,∗ , A.A. Tovsultanov b a b

Peoples’ Friendship University of Russia, Miklukho-Maklaya 6, 117198 Moscow, Russia Chechen State University, Sheripova 32, 364024 Grozny, Russia

a r t i c l e

i n f o

Article history: Received 25 April 2018 Available online 12 August 2019 Submitted by J. Shi Keywords: Elliptic functional differential equation Boundary value problem Differential-difference equation Rescaling

a b s t r a c t We study the Dirichlet problem for a functional differential equation containing shifted and contracted argument under the Laplacian sign. We establish conditions for the unique solvability and demonstrate also that the problem may have an infinite dimensional solution manifold. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Functional differential equations containing affine transformations of the argument, i.e. shifts combined with contractions (expansions), generalize the well-known pantograph equation and have applications in a wide variety of fields such as astrophysics [1], nonlinear oscillations [11], biology [7], number theory [10], and probability theory [6]. They may be considered as a pattern for equations with unlimited delay. On the other hand, the study of their multidimensional analogs is essential in constructing a general theory of elliptic boundary value problems associated with an infinite non-isometric group of transformations. Functional differential equations with affine transformations on the line and in the complex plane have been studied in detail since the 1970s in connection with the existence of bounded solutions and the asymptotic behavior of solutions at infinity, see, for example, [9,5,2,4,8] (to date, a considerable number of papers have been published by the same authors as well as some other mathematicians). The boundary value problem for a functional differential equation with the contracted one-dimensional argument was considered for the first time in [3]. The present state of the theory of boundary value problems ✩ The publication has been prepared with the support of the “RUDN University Program 5-100” and by the Russian Foundation for Basic Research (project no. 17-01-00401). * Corresponding author. E-mail addresses: [email protected] (L.E. Rossovskii), [email protected] (A.A. Tovsultanov).

https://doi.org/10.1016/j.jmaa.2019.123403 0022-247X/© 2019 Elsevier Inc. All rights reserved.

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for elliptic functional differential equations is reflected in [15–17,12,13], where the problems were studied either in the presence of solely shifts (differential-difference equations) or contractions (expansions). In this paper, we consider a model equation (which means that a single functional operator appears under the Laplacian sign) containing both shift and contractions. The difficulties that arise in this case are connected with the fact that the contraction and the shift are noncommutative transformations. In addition, the simultaneous presence of these types of transformations in the equation leads to the appearance of an infinite number of fixed points. The problem is considered under the assumption that all these fixed points belong to the closure of the domain. We use the customary Sobolev spaces H s (Rn ) and H s (Ω) for s ∈ R, H s (Rn ) being the of  Hilbert space  all tempered distributions u in Rn such that their Fourier transforms u ˜(ξ) belong to L2 Rnξ ; (1 + |ξ|2 )s dξ , and H s (Ω) the space of restrictions of u ∈ H s (Rn ) to Ω ⊂ Rn . 2. Properties of functional operators with affine transformations of the argument Let K be a compact set in Rn and ν ∈ (C(K))∗ a regular complex-valued Borel measure concentrated on K. Assuming that a function u is defined in Rn , consider the convolution operator ˆ (ν ∗ u)(x) =

u(x − h) dν(h). K

Clearly, if u ∈ C0 (Rn ), then the convolution is a compactly supported continuous function as well with supp (ν ∗ u) ⊂ supp u + K. If u ∈ C0∞ (Rn ), then (ν ∗ u) ∈ C0∞ (Rn ). Indeed, fixed any point x, the function family t−1 (u(x + tej − h) − u(x − h)) converges to uxj (x − h) as t → 0 uniformly with respect to h ∈ K, therefore, one can pass to the limit under the integral sign to obtain (ν ∗ u)xj = ν ∗ uxj .

(1)

Applying the Fourier transform, we easily verify with the help of Fubini’s theorem that  (ν ∗ u)(ξ) = ν˜(ξ)˜ u(ξ), where ν˜(ξ) =

ˆ

e−ihξ dν(h)

K

is the characteristic function of the measure ν, uniformly continuous and bounded in Rn (being the restriction to Rn of an entire function in C n , see [14, Theorem 7.23]). Obviously, ν˜(0) = ν(K) and sup |˜ ν (ξ)|  |ν|(K), where |ν| is the variation of the measure ν. Thus, the convolution uniquely extends to a bounded linear operator in the space L2 (Rn ) and the Sobolev spaces H s (Rn ), s ∈ R, with the same norm equal to sup |˜ ν (ξ)|, while relation (1) now holds for generalized derivatives. Iterated convolution with measures ν1 ∈ (C(K1 ))∗ and ν2 ∈ (C(K2 ))∗ , being multiplication by the function ν˜1 (ξ)˜ ν2 (ξ) for the Fourier images, is convolution with a measure ν1 ∗ ν2 , (ν1 ∗ (ν2 ∗ u)) = (ν1 ∗ ν2 ) ∗ u, supported in K1 + K2 and defined by the formula ˆ (ν1 ∗ ν2 )(B) =

ˆ ν2 (B − h) dν1 (h) =

K1

ν1 (B − h) dν2 (h) K2

(here B is any Borel set in Rn ). The norm of such operator (both in L2 and H s ) equals sup |˜ ν1 (ξ)˜ ν2 (ξ)|. Given a measure ν ∈ (C(K))∗ , consider the measure ν • ∈ (C(−K))∗ defined by ν • (B) = ν(−B). It is easily seen that ν• = ν˜. Convolution with the measure ν • is the adjoint operator to convolution with ν in the space L2 (Rn ).

L.E. Rossovskii, A.A. Tovsultanov / J. Math. Anal. Appl. 480 (2019) 123403

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Note that if Ω and Ω are open subsets in Rn such that Ω − K ⊂ Ω, then the restriction of the function ν ∗ u to Ω only depends on the restriction of a function u to Ω, therefore, convolution with a measure ν ∈ (C(K))∗ also acts as a bounded linear operator from L2 (Ω) to L2 (Ω ), and from H s (Ω) to H s (Ω ) (with norm not exceeding sup |˜ ν (ξ)| if the norm of a function from H s (Ω) is understood as the infimum of s n the norms in H (R ) for all admissible extensions of this function to Rn ). A generalized derivative of the function ν ∗ u ∈ H s (Ω ) is still connected with that one of u ∈ H s (Ω) by formula (1). Fixed a number p > 1, denote P u(x) = u(p−1 x). An operator with affine transformations of the argument has the form ˆ T u(x) = P (ν ∗ u)(x) = u(p−1 x − h) dν(h). (2) K

Remark 1. If a measure ν is supported within a finite set K = {h1 , . . . , hl } (the case of an atomic measure), then this operator is indeed an operator with affine transformations of the argument, i.e. combinations of shifts and contractions, P (ν ∗ u)(x) = α1 u(p−1 x − h1 ) + . . . + αl u(p−1 x − hl ) (α1 , . . . , αl ∈ C). We keep this term for more general measures spread over some compact set. We can write down the operator T in a different way, interchanging convolution and contraction, ˆ T u(x) =

u(p−1 (x − ph)) dν(h) =

K

ˆ

u(p−1 (x − h)) dP ν(h) = (P ν ∗ P u)(x).

pK

Here P ν is a measure on pK defined by P ν(B) = ν(p−1 B). Lemma 1. The spectral radius of the operator T : L2 (Rn ) → L2 (Rn ) is calculated by the formula ρ(T ) = pn/2 lim sup |˜ ν (ξ)˜ ν (p−1 ξ) . . . ν˜(p1−m ξ)|1/m m→∞ ξ∈Rn

= pn/2 lim sup |˜ ν (ξ)˜ ν (pξ) . . . ν˜(pm−1 ξ)|1/m . m→∞ ξ∈Rn

(3)

Proof. The Fourier image of the function T u is P ∗ (˜ νu ˜)(ξ) = pn ν˜(pξ)˜ u(pξ). Therefore, ˆ T u2L2 (Rn ) = p2n

ˆ |˜ ν (pξ)˜ u(pξ)|2 dξ = pn

Rn

|˜ ν (η)˜ u(η)|2 dη,

Rn

and it is immediately seen that the norm of the operator T acting in L2 (Rn ) equals pn/2 sup |˜ ν (ξ)|. For powers m = 2, 3, . . . of the operator T , we similarly have T m u = P m (P 1−m ν ∗ . . . ∗ P −1 ν ∗ ν ∗ u), m u)(ξ) = pmn ν  ˜(pξ)˜ ν (p2 ξ) . . . ν˜(pm ξ)˜ u(pm ξ), (T ˆ ν (η)˜ ν (p−1 η) . . . ν˜(p1−m η)|2 |˜ u(η)|2 dη, T m u2L2 (Rn ) = pmn |˜ Rn

T

m

: L2 (R ) → L2 (Rn ) = pmn/2 sup |˜ ν (ξ)˜ ν (p−1 ξ) . . . ν˜(p1−m ξ)|. n

In view of the well-known formula for the spectral radius [14, Theorem 10.13], this implies (3).

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Example 1. Let 0 = h0 ∈ Rn , K = {h0 , −h0 }, a, b ∈ R, and δ denote the Dirac measure (the delta function). Take ν = aδ(h + h0 ) + bδ(h − h0 ). We have T u(x) = au(p−1 x + h0 ) + bu(p−1 x − h0 ), 0

ν˜(ξ) = aeih

ξ

+ be−ih

0

= (a + b) cos(h0 ξ) + i(a − b) sin(h0 ξ),

ξ

|˜ ν (ξ)|2 = (a + b)2 cos2 (h0 ξ) + (a − b)2 sin2 (h0 ξ) = (a2 + b2 )(1 + κ cos(2h0 ξ)), κ = 2ab/(a2 + b2 ),

|κ|  1,

and, according to (3), ρ(T ) = = pn/2 (a2 + b2 )1/2 lim sup |(1 + κ cos t)(1 + κ cos pt) . . . (1 + κ cos pm−1 t)|1/2m . m→∞ t∈R

The cases where κ > 0 and κ < 0 lead to different conclusions. If κ > 0 (a and b are of the same sign), then the supremum occurring here is attained for t = 0, it is the same for all m and equals (1 + κ)1/2 . Thus, the spectral radius coincides with the norm of the operator T , ρ(T ) = pn/2 (|a| + |b|). If κ < 0 (a and b have opposite signs), then the situation depends essentially on p. Let, for example, a = 1 and b = −1. We can write ρ(T ) = 2pn/2 lim sup | sin t sin pt . . . sin pm−1 t|1/m . m→∞ t∈R

When p is an odd integer, | sin(π/2) sin(pπ/2) . . . sin(pm−1 π/2)| = 1. Here again we come to a stationary sequence, and the spectral radius is 2pn/2 , which is the norm of T . But if, for example, p = 2, then, as m grows, the corresponding sequence is no longer stationary, and ρ(T ) < 2pn/2 . So, we have sup | sin t sin 2t|1/2 = 2 · 3−3/4 < 1, and sup | sin t sin 2t sin 4t|1/3 is even less, etc. Example 2. Let K = {x ∈ R3 : |x| = r} be the sphere of radius r in R3 and ν the normalized area on the sphere. Convolution now averages over the sphere, 1 T u(x) = 4πr2

¨

u(p−1 x − h) dSh ,

|h|=r

and it is readily seen that 1 ν˜(ξ) = 4πr2

¨

e−iξh dSh =

|x|=r

sin r|ξ| , r|ξ|

1/m   sin t sin pt sin pm−1 t  3/2  ρ(T ) = p . . . m−1  lim sup = p3/2 . m→∞ t>0  t pt p t In the examples above, the value of ρ(T ) does not depend on the shift size. Let us turn to operators in a bounded domain Ω. Here the principal condition p−1 Ω − K ⊂ Ω is imposed on the couple Ω, K.

(4)

L.E. Rossovskii, A.A. Tovsultanov / J. Math. Anal. Appl. 480 (2019) 123403

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Remark 2. If K consists of only one point h, then condition (4) can be interpreted as follows. The transformation x → p−1 x − h is contraction of Rn with center x0 = ph/(1 − p), and, starting from any point x1 , the iteration sequence xk+1 = p−1 xk − h converges to x0 . Take x1 ∈ Ω, then condition (4) ensures xk ∈ Ω, k = 2, 3, . . ., and it follows that x0 ∈ Ω. Thus, the center of contraction belongs to the closure of the domain, and the contracted domain lies in the original one. Due to the arguments above, integral (2) may be viewed as contraction with centers distributed over Ω. Condition (4) allows one to also consider T as a bounded operator in L2 (Ω). This operator is a composition of the convolution operator acting from L2 (Ω) to L2 (p−1 Ω), and the contraction operator P from L2 (p−1 Ω) to L2 (Ω). The same applies to the space H s (Ω). Lemma 2. Suppose (4) holds and a number α ∈ C is such that |α| < 1/ρ(T ). Then the operator I + αT : H s (Ω) → H s (Ω) has a bounded inverse for all s  0. If, in addition, α satisfies the stronger inequality |α| < 1/(ps ρ(T )) for some positive s, then the operator I + αT : H −s (Ω) → H −s (Ω) is boundedly invertible as well. Proof. Under lemma’s hypothesis, the operator I + αT : L2 (Rn ) → L2 (Rn ) has the bounded inverse (I + αT )−1 v =

∞  m=0

(−α)m T m v =

∞ 

(−α)m P m (νm ∗ v),

(5)

m=0

νm = P 1−m ν ∗ . . . ∗ P −1 ν ∗ ν. Let s  0 and v ∈ H s (Rn ). Then the term of the series admits the estimate ˆ = (pn |α|)2m ˆ

(−α)m T m v2H s (Rn ) |˜ ν (pξ)˜ ν (p2 ξ) . . . ν˜(pm ξ)|2 (1 + |ξ|2 )s |˜ v (pm ξ)|2 dξ

Rn

= (pn |α|2 )m

|˜ ν (η)˜ ν (p−1 η) . . . ν˜(p1−m η)|2 (1 + |p−m η|2 )s |˜ v (η)|2 dη

Rn

 (pn |α|2 )m sup |˜ ν (η)˜ ν (p−1 η) . . . ν˜(p1−m η)|2 η∈Rn

ˆ (1 + |η|2 )s |˜ v (η)|2 dη.

Rn

´ The coefficient of v2H s (Rn ) = Rn (1 + |η|2 )s |˜ v (η)|2 dη is majorized by a decreasing geometric progression, s therefore, operator (5) is also bounded in H (Rn ). Take any function v ∈ H −s (Rn ). Since (1 + |p−m η|2 )−s  p2ms (1 + |η|2 )−s , we have the estimate (−α)m T m v2H −s (Rn )   (pn+2s |α|2 )m sup |˜ ν (η)˜ ν (p−1 η) . . . ν˜(p1−m η)|2 v2H −s (Rn ) . η∈Rn

The uniform convergence of the series is guaranteed now by the inequality |α| < 1/(ps ρ(T )). Now turn to the bounded domain Ω. The measure νm is supported within the compact set Km = K + −1 p K+. . . p1−m K. It follows from (4) that p−2 Ω−p−1 K ⊂ p−1 Ω and thus p−2 Ω−p−1 K−K ⊂ p−1 Ω−K ⊂ Ω, implying p−2 Ω − K2 ⊂ Ω. Continuing in a similar way, we come to the inclusion p−m Ω − Km ⊂ Ω. This means that the restriction of the function P m (νm ∗ v) to Ω is uniquely determined by the restriction of a function v to Ω. Thus, formula (5) defines also a bounded operator in H ±s (Ω) being inverse to the operator I + αT : H ±s (Ω) → H ±s (Ω).

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3. Solvability of the boundary value problem Under the conditions of the previous paragraph, consider the boundary value problem −Δ(u(x) + αP (ν ∗ u)(x)) = f (x) (x ∈ Ω),

(6)

u|∂Ω = 0.

(7)

Assuming f ∈ L2 (Ω), we introduce a generalized solution to problem (6) and (7) in a usual way as a function ˚1 (Ω) satisfying the integral identity u∈H n  ((u + αP (ν ∗ u))xj , vxj )L2 (Ω) = (f, v)L2 (Ω)

(8)

j=1

˚1 (Ω). for all test functions v ∈ H Theorem 3. Let geometric condition (4) and the inequality  |α| < p1−n/2

−1 ν (ξ)˜ ν (pξ) . . . ν˜(pm−1 ξ)|1/m lim sup |˜

m→∞ ξ∈Rn

(9)

˚1 (Ω) for any function f ∈ L2 (Ω). hold. Then problem (6) and (7) has a unique generalized solution u ∈ H If, in addition, f ∈ H k (Ω) and ∂Ω ∈ C k+2 (k nonnegative integer), then u ∈ H k+2 (Ω). ˚ 1 (Ω) onto the Proof. It is well known that the Laplacian acts as a linear homeomorphism of the space H 1 ∗ −1 ˚ (Ω)) = H (Ω). Rewriting the expression on the left-hand side in (6) as adjoint space (H −(Δu + αp−2 P (ν ∗ Δu)) = −(I + αp−2 T )Δu with Δu ∈ H −1 (Ω), we see that the question of unique solvability of problem (6) and (7) becomes the question of invertibility of the operator I + αp−2 T in the space H −1 (Ω). Inequality (9) implies |α|p−2 < 1/(pρ(T )), and application of Lemma 2 proves the existence of the bounded inverse operator (I +αp−2 T )−1 : H −1 (Ω) → H −1 (Ω). Thus, under the hypothesis of the theorem, the original problem is equivalent to the Dirichlet problem for −Δu = (I + αp−2 T )−1 f . Now the assertion follows from the well-known properties of the Dirichlet problem for the Poisson equation and from the boundedness of the operator (I + αp−2 T )−1 also in the spaces L2 (Ω) and H k (Ω). Remark 3. Of course, problem (6) and (7) has a unique generalized solution also in the case where f ∈ H −1 (Ω) as soon as the expression on the right-hand side in (8) is understood as the action of the distribution ˚1 (Ω). f on the test function v ∈ H Remark 4. Transition from problem (6) and (7) to the Dirichlet problem for the Poisson equation −Δu = (I + αp−2 T )−1 f can be demonstrated based on integral identity (8). ˚1 (Ω), the functions ν • ∗ (P −m v) belong to H ˚1 (Ω) as well, and can replace v in (8): For a function v ∈ H m n 

• • ((I + αp−1 T )uxj , pm νm ∗ (P −m vxj ))L2 (Ω) = (f, νm ∗ (P −m v))L2 (Ω) ,

j=1 n  (pm P m (νm ∗ (I + αp−1 T )uxj ), vxj )L2 (Ω) = (P m (νm ∗ f ), v)L2 (Ω) . j=1

L.E. Rossovskii, A.A. Tovsultanov / J. Math. Anal. Appl. 480 (2019) 123403

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Summing these equations, multiplied by (−αp−2 )m , over all m = 0, 1, . . ., we get n  (uxj , vxj )L2 (Ω) = ((I + αp−2 T )−1 f, v))L2 (Ω) j=1

with the help of Lemma 2. Example 3. Consider the Dirichlet problem for the equation ⎛ ⎜ −Δ ⎝u(x) + α

⎞

¨

⎟ u(p−1 x − h) dSh ⎠ = f (x)

(10)

|h|=r

in the ball Ω = {x ∈ R3 : |x| < R}, where r < (p − 1)R/p. The latter inequality ensures the fulfillment of (4). Theorem 3 together with Example 2 show the unique solvability of problem (10) and (7) provided 4πp1/2 r2 |α| < 1. Example 4. We give an example showing that for large absolute values of α problem (6) and (7) may have infinitely many generalized solutions. Let Ω ⊂ R2 be the square {−1 < x1 , x2 < 1} and h = (1, 1). Consider the Dirichlet problem for the equation −Δ (u(x) + α[u((x + h)/2) − u((x − h)/2))]) = f (x) (x ∈ Ω).

(11)

Here T u(x) = u((x + h)/2) − u((x − h)/2)), condition (4) is satisfied (the operator is a combination of contractions with centers at the vertices (1, 1) and (−1, −1) of the square), and the whole expression under ˚1 (Ω) → H 1 (Ω). the Laplacian sign defines the bounded operator I + αT : H ˚1 (Ω) for any function w ∈ H ˚1 (Ω) Show that the equation u +αT u = w has infinitely many solutions u ∈ H (in the sense that these solutions form an infinite-dimensional linear manifold) if |α| > 1. Then boundary value problem (11), (7) has infinitely many generalized solutions for any function f ∈ L2 (Ω) as well. On the other hand, Theorem 3 guarantees unique solvability of (11), (7) for |α|  1/2 (and even for |α|  33/4 /4, the latter assessment can be further improved, see Example 1). Fix w and construct some solutions to u + αT u = w. We need the following notation for that: Ω1 = {0 < x1 , x2 < 1},

Ω2 = {−1 < x1 < 0, 0 < x2 < 1},

Ω3 = {−1 < x1 , x2 < 0},

Ω4 = {0 < x1 < 1, −1 < x2 < 0},

Ω11 = {1/2 < x1 , x2 < 1},

Ω12 = {0 < x1 < 1/2, 1/2 < x2 < 1},

Ω13 = {0 < x1 , x2 < 1/2},

Ω14 = {1/2 < x1 < 1, 0 < x2 < 1/2}

(these are the parts of the square Ω in the coordinate quarters, where the part Ω1 lying in the first quarter is divided in its turn into four parts), Γ1 = {(x1 , −1/2) : −1 < x1 < 0},

Γ2 = {(−1/2, x2 ) : −1 < x2 < 0}.

We also denote by ui the restrictions of a sought function u to the squares Ωi , and by u1i to the squares ˚1 (Ωi ), u1i ∈ H ˚1 (Ω1i ). Each of the functions u, ui , u1i is supposed to be Ω1i , i = 1, 2, 3, 4. We seek ui ∈ H extended by zero outside of the corresponding square, so there is no misunderstanding in the records u = u1 + u2 + u3 + u4 ,

u1 = u11 + u12 + u13 + u14 .

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L.E. Rossovskii, A.A. Tovsultanov / J. Math. Anal. Appl. 480 (2019) 123403

Taking all this into account, we may rewrite the functional equation under study in the form of a system, u1 (x) + αu1 ((x + h)/2) − αu3 ((x − h)/2) = w(x) (x ∈ Ω1 ),

(12)

u2 (x) + αu1 ((x + h)/2) − αu3 ((x − h)/2) = w(x) (x ∈ Ω2 ),

(13)

u3 (x) + αu1 ((x + h)/2) − αu3 ((x − h)/2) = w(x) (x ∈ Ω3 ),

(14)

u4 (x) + αu1 ((x + h)/2) − αu3 ((x − h)/2) = w(x) (x ∈ Ω4 ).

(15)

˚1 (Ω3 ) satisfying the conditions We set a function u3 ∈ H u3 |Γ1 = −α−1 w(2x + h)|Γ1 ,

u3 |Γ2 = −α−1 w(2x + h)|Γ2 ,

(16)

while completely arbitrary in other respects. Then the values of u1 in Ω13 are prescribed by Equation (14), u13 (x) = u1 (x) = α−1 (w(2x − h) − u3 (2x − h)) + u3 (x − h)

(x ∈ Ω13 ),

˚1 (Ω13 ), by conditions (16) imposed on u3 . where the function u13 has the zero trace on ∂Ω13 , i.e. u13 ∈ H Indeed, taking, for example, the side 0 < x1 < 1/2, x2 = 1/2 of the square, we have u13 (x1 , 1/2) = α−1 (w(2x1 − 1, 0) − u3 (2x1 − 1, 0)) + u3 (x1 − 1, −1/2) = α−1 w(2x1 − 1, 0) + u3 (x1 − 1, −1/2) = 0. Put u12 = 0, u14 = 0, and turn to equation (12) from which we are going to find u11 . Rewrite this equation as follows: u11 (x) + αu11 ((x + h)/2) = w(x) + αu3 ((x − h)/2) − u13 (x) (x ∈ Ω1 ).

(17)

˚ 1 (Ω1 ) by virtue of conditions (16), while the expression on the Here the function on the right belongs to H left is a sum of the identity operator and an operator where the argument undergo contraction with respect to the point (1, 1). It follows from [12, Lemma 1.2] that for |α| > 1 the operator on the left-hand side is ˚ 1 (Ω11 ) onto H ˚1 (Ω1 ). Thus, equation (17) uniquely defines u11 ∈ H ˚1 (Ω11 ), a linear homeomorphism of H and the construction of u1 is complete. After this, the components u2 and u4 can simply be found from ˚1 (Ω2 ) and H ˚1 (Ω4 ), (13) and (15), where again conditions (16) guarantee that these components belong to H respectively. References [1] V.A. Ambartsumyan, On the theory of brightness fluctuations in the Milky Way, Dokl. Akad. Nauk SSSR 44 (1944) 244–247. [2] L.V. Bogachev, G.A. Derfel’, S.A. Molchanov, On bounded continuous solutions of the archetypal equation with rescaling, Proc. R. Soc. A 471 (2015), https://doi.org/10.1098/rspa.2015.0351. [3] K. Cooke, L.E. Rossovskii, A.L. Skubachevskii, A boundary value problem for a functional-differential equation with a linearly transformed argument, Differ. Equ. 31 (1995) 1294–1299. [4] G. Derfel’, B. van Brunt, G.C. Wake, A cell growth model revisited, Funct. Differ. Equ. 19 (2012) 71–81. [5] G.A. Derfel’, S.A. Molchanov, Spectral methods in the theory of differential-functional equations, Math. Notes 47 (1990) 42–51. [6] D.P. Gaver Jr., An absorption probability problem, J. Math. Anal. Appl. 9 (1964) 384–393. [7] A.J. Hall, G.C. Wake, A functional differential equation arising in the modeling of cell growth, J. Aust. Math. Soc. Ser. B 30 (1989) 424–435. [8] A. Iserles, On neutral functional-differential equation with proportional delays, J. Math. Anal. Appl. 207 (1997) 73–95. [9] T. Kato, J.B. McLeod, Functional differential equation y˙ = ay(λt) + by(t), Bull. Amer. Math. Soc. 77 (1971) 891–937. [10] K. Mahler, On a special functional equation, J. Lond. Math. Soc. 15 (1940) 115–123. [11] J.R. Ockendon, A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R. Soc. Lond. A 322 (1971) 447–468.

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[12] L.E. Rossovskii, Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function, J. Math. Sci. 223 (2017) 351–493. [13] L.E. Rossovskii, Elliptic functional differential equations with incommensurable contractions, Math. Model. Nat. Phenom. 12 (2017) 226–239. [14] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1991. [15] A.L. Skubachevskii, Elliptic Functional-Differential Equations and Applications, Oper. Theory Adv. Appl., vol. 91, Birkhäuser Verlag, Basel, 1997. [16] A.L. Skubachevskii, Boundary-value problems for elliptic functional-differential equations and their applications, Russian Math. Surveys 71 (2016) 801–906. [17] A.L. Skubachevskii, Nonlocal problems in the mechanics of three-layer shells, Math. Model. Nat. Phenom. 12 (2017) 192–207.